A variational approach to the alternating projections method

摘要

The 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets {A(n)} and {B-n}, each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point a(0), we consider the sequences of points obtained by projecting on the "perturbed" sets, i.e., the sequences {a(n)} and {b(n)} given by b(n) = P-Bn(a(n-1)) and a(n)=PA(n)(b(n)). Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences {a(n)} and {b(n)} converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection A boolean AND B reduces to a singleton and when the interior of A boolean AND B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.